The noun is Sponsor and not Sponser, so why is it sponsern and not sponsorn? Is this because Sponsor is an anglicism? Are there other examples of such strange flexions in the German language?
Short answer to your question: yes this is because sponsern is an anglicism.
See also Korrekturen.de on this.
As for your second question, whether there are similar examples, I’d say pretty much every word that is taken from English is a potential candidate. This depends on how “correctly” the verb was taken into the German language.
An example: the verb to download is by now also widely used in Germany in the form etwas downloaden. As you can see, the usual verb ending -en was attached. Now in the case of sponsern, it seems that people found it more convenient to use sponsern instead of the correct form sponsoren. However, the form sponsoren also exists.
So the verb sponsern is merely an abbreviation of the correct form sponsoren but over the years this, initially incorrect, version found its ways into German dictionaries.
Suppose we have a word Sponsor/to sponsor (the German noun or the English verb; it does not matter) and want to derive a German verb from it.
Let's have a little look at the history of the word sponsor. It is a nomen agentis derived from the latin verb spondere. Nomina agentis can be formed by suffixing -tor to the verbal root (spond-). Two contacting dental plosives (i.e. "d" and "t") yielded an "s"; so the nomen agentis ending becomes -sor and we have sponsor containing the modified root spons-.
Potentially, the German verb can be derived from the roots spons- or spond- or the full word sponsor. The average German speaker cannot know the above etymology. Thus, if the verb spondere is not loaned parallelly into German he/she cannot know that there was a "d" in the root. The German verb we want to derive must contain an "s". Now there are several possibilities to form a verb: The verb must end in -en, -ern or -eln; the verb may end in -ieren but it need not. The -eln can be excluded because there is no "l" in the last syllable of sponsor.
- If the "d" of the root were known, it would be possible to form a verb spondieren. Actually, there is such a verb and it is related to latin spondere. However, it has a totally different meaning from "to sponsor" and does not come from English but directly from Latin into German.
- Another possibility is sponsieren; again, this verb exists - but again with a totally different meaning.
- As somebody mentioned before, the verb sponsoren exists, too. It is derived from English to sponsor.
- Finally, another possibility is sponsern, the main variant of the verb that is derived from English to sponsor.
Now, why did sponsern win and sponsoren lose, i.e. why is sponsern so common? - The answer lies in German word stress: German prefers Trochee words. Those are words with the main stress on the first (root) syllable and an unstressed Schwa-sound in the second (inflection suffix) syllable. Trochee is the typical German meter; most German nouns/verbs have two syllables and follow this structure (e.g. die Wolke, die Wolken) and often nouns with only one syllable form plurals with a Trochee structure (e.g. die Frau, die Frauen). sponsern follows the Trochee structure, sponsoren does not; that's why sponsern won.
N.B.: The reason why sponsern dominates sponsoren must lie in German itself and not in the fact that sponsern came from English. This gets clear if you consider the verb herumdoktern that is derived from the noun Doktor and not a loanword. -doktern follows the Trochee principle, -doktoren would not.
The ending -or is used to indicate a person or a thing doing something. A Sponsor, Prediktor + Korrektor in mathematics, Indikator, Rektor, Reaktor and many others. The Prediktor doesn't predictor, it predicts. The Indikator doesn't indicator, it indicates. So when turning a noun ending in -or into a verb, it is quite natural to remove the -or.